Assume not. Then \(q^\nabla \)  has image in \(\operatorname{At}_{(C,D)}(E_\star , P^\bullet )\) , and
hence yields a splitting of (REF ).
Using proposition:atiyah-splittings,
the corresponding connection with regular singularities on \(E\)  preserves \(P^1\)  and hence yields a flat connection on \(P^1\)  with regular singularities along
\(D\) , whose monodromy is a sub-representation of the monodromy representation
\(\rho \)  associated to \(({E}, \nabla )|_{C\setminus D}\) .
But this contradicts the
assumption that \(\rho \)  is irreducible. (See [1]} for a similar argument in the non-parabolic setting.)
